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Unsteady MagnetohydrodynamicConvective Boundary Layer Flow past aSphere in Viscous and Micropolar Fluids

Nurul Farahain Mohammad

Department of Mathematical Sciences, Faculty of Science,Universiti Teknologi Malaysia

Name of supervisors:Assoc. Prof. Dr. Sharidan Shafie and Dr. Anati Ali.

1

Outline of Presentation

Introduction

Problem 1 / Chapter 4

Problem 2 / Chapter 5

Problem 3 / Chapter 6

Problem 4 / Chapter 7

Problem 5 / Chapter 8

Conclusion

Suggestions for Future Works

Publication / Awards / Attended Conferences

2

Introduction

Objectives

to examine the effects of MHD on unsteady boundary layerflow over a sphere.

to analyze the behaviour of viscous fluid and micropolarfluid under the influence of MHD.

to investigate the interaction between MHD flow and heattransfer with or without buoyancy force.

3

Introduction

Scope

electrically-conducting viscous and micropolar fluids incompressible unsteady 2D laminar boundary layer flow

past a sphere uniform magnetic field, transverse of the fluid flow induced magnetic field is neglected no polarized or applied voltage enforced on the fluid flow numerical solution (Keller-Box method) no real experiments conducted to validate the numerical

results

4

Research Methodology

Mathematical Analysis

dimensionless variables stream function similarity transformation

Keller-Box Method

finite difference method Newtons method block-tridiagonal factorization scheme

5

Problems Solved

O

a

U

x/ayr(x)

x

Tw

T

T

g

Figure : Physical Coordinate

Viscous fluid Micropolar fluidsBoundary Layer Flow Prob. 1 / Chap. 4 Prob. 4 / Chap. 7Forced Convection Prob. 2 / Chap. 5 -Mixed Convection Prob. 3 / Chap. 6 Prob. 5 / Chap. 8

6

Problem 1 / Chapter 4

(r (x) u

)x

+(r (x) v

)y

= 0, (1)

ut

+ uux

+ vuy

= 1

px

+

(2ux2

+2uy2

) B0

2

u, (2)

vt

+ uvx

+ vvy

= 1

py

+

(2vx2

+2vy2

) B0

2

v , (3)

subject to the following initial and boundary conditions:

t < 0 : u = v = 0, for any x , y ,

t 0 : u = v = 0, at y = 0,u = ue(x), as y .

(4)

7

Dimensional Governing Equations

Non-dimensional Governing Equations

Governing Equations in Stream Function

Non-similar Governing Equations

Dimensionless variables

Stream function

Similarity variables

8

Discretized Governing Equations

Linearized Numerical Scheme

Block Tridiagonal Factorization Scheme

Equations solved

ts, xs, f ,Cf Re1/2,h, s,NuRe1/2

Finite Difference Method

Newtons Method

LU factorization

Block Elimination Method

Analyse

9

Problem 1 / Chapter 4

Table : The separation times of flow past the surface of a sphere.

x M = 0 M = 0 M = 0.1 M = 0.5 M = 1.0 M = 1.3(Ali, 2010) (present)

180 0.3966 0.3960 0.4161 0.5241 0.7963 1.2470171 0.4016 0.4010 0.4217 0.5331 0.8186 1.3103162 0.4177 0.4170 0.4394 0.5623 0.8940 1.5677153 0.4471 0.4463 0.4721 0.6178 1.0627 -144 0.4947 0.4937 0.5257 0.7152 1.4428 -135 0.5709 0.5694 0.6128 0.8937 - -126 0.6987 0.6960 0.7632 1.2953 - -117 0.9442 0.9372 1.0779 - - -108 - 1.6751 - - - -

Ali, A. (2010). Unsteady micropolar boundary layer flow and convective heattransfer. Universiti Teknologi Malaysia, Faculty of Science: PhD Thesis.

10

Problem 1 / Chapter 4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f

M = 0, 0.1, 0.5, 1.0, 1.5

0 10 20 30 40 50 60 70 80 900.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

f

M = 0, 0.1, 0.5, 1.0, 1.5

x = 0 x = 180

11

Problem 1 / Chapter 4

0 20 40 60 80 100 120 140 160 1801.5

1

0.5

0

0.5

1

1.5

2

2.5

3

x

Cf R

e1/2

t = 0.1, 0.5, 1.0, 1.5, 2.0

0 20 40 60 80 100 120 140 160 1800

0.5

1

1.5

2

2.5

3

3.5

x

Cf R

e1/2

= 0.1, 0.5, 1.0, 1.5, 2.0 t

Without MHD MHD

12

Problem 2 / Chapter 5

(r (x) u

)x

+(r (x) v

)y

= 0, (5)

ut

+ uux

+ vuy

= 1

px

+

(2ux2

+2uy2

) B0

2

u, (6)

vt

+ uvx

+ vvy

= 1

py

+

(2vx2

+2vy2

) B0

2

v , (7)

Cp(Tt

+ uTx

+ vTy

)= c

(2Tx2

+2Ty2

), (8)

subject to the following initial and boundary conditions:

t < 0 : u = v = 0,T = T for any x , y ,

t 0 : u = v = 0,T = Tw at y = 0,

u = ue(x),T = T as y .

(9)

13

Problem 2 / Chapter 5

Ms NuRe1/2

xCf(x = 0) (x = 180) (x = 0) (x = 180)

Pr s NuRe1/2

t NuRe1/2

14

Problem 3 / Chapter 6 (r u)x

+ (r v)y

= 0, (10)

(ut

+ uux

+ vuy

)= p

x+

(2ux2

+2uy2

) B20u g(T T) sin

(xa

),

(11)

(vt

+ uvx

+ vvy

)= p

y+

(2vx2

+2vy2

) B20v + g(T T) cos

(xa

),

(12)

Cp(Tt

+ uTx

+ vTy

)= c

(2Tx2

+2Ty2

), (13)

subject to the following initial and boundary conditions:

t < 0 : u = v = 0,T = T for any x , y ,

t 0 : u = v = 0,T = Tw at y = 0,

u = ue(x),T = T as y .

(14)

15

Problem 3 / Chapter 6

M ts xs Cf Re1/2NuRe1/2

(x = 0) (x = 180)

ts xs Cf Re1/2NuRe1/2

(x = 0) (x = 180)

16

Problem 4 / Chapter 7(r (x) u

)x

+(r (x) v

)y

= 0, (15)

(ut

+ uux

+ vuy

)= p

x+(+)

(2ux2

+2uy2

)+

NyB20u, (16)

(vt

+ uvx

+ vvy

)= p

y+(+)

(2vx2

+2vy2

)N

xB20v , (17)

j(Nt

+ uNx

+ vNy

)=

(2Nx2

+2Ny2

)

(2N +

uy vx

), (18)

subject to the following initial and boundary conditions:

t < 0 : u = v = N = 0, for any x , y ,

t 0 : u = v = 0,N = nuy, at y = 0,

u = ue(x),N = 0, as y .

(19)

17

Problem 4 / Chapter 7

M ts xs f Cf Re1/2h

(x = 0) (x = 180)

( = 0) ( = 0, = ) ( ) ( )

K ts f h

18

Problem 5 / Chapter 8(r (x) u

)x

+(r (x) v

)y

= 0, (20)

(ut

+ uux

+ vuy

)= p

x+ (+ )

(2ux2

+2uy2

)+

Ny B20u + g

(T T

)sin x ,

(21)

(vt

+ uvx

+ vvy

)= p

y+ (+ )

(2vx2

+2vy2

) N

x B20v g

(T T

)cos x ,

(22)

j(Nt

+ uNx

+ vNy

)=

(2Nx2

+2Ny2

)

(2N +

uy vx

), (23)

Cp(Tt

+ uTx

+ vTy

)= c

(2Tx2

+2Ty2

), (24)

19

Problem 5 / Chapter 8

subject to the following initial and boundary conditions:

t < 0 : u = v = N = 0,T = T for any x , y ,

t 0 : u = v = 0,N = nuy,T = Tw at y = 0,

u = ue(x),N = 0,T = T as y .

(25)

20

Problem 5 / Chapter 8

M ts xs Cf Re1/2NuRe1/2

(x = 0) (x = 180)

(Pr = 0.7) (Pr = 7)

K = 1,n = 0 = 1 = 1Pr = 0.7 M = 0.9 M = 2Pr = 7 M = 1.2 M = 1.5

K = 1,n = 0.5 = 1 = 1Pr = 0.7 M = 0.9 M = 2Pr = 7 M = 1.2 M = 1.7

21

Conclusion 5 different unsteady MHD models in viscous and

micropolar fluids over a sphere. 3-dimensional numerical schemes. MATLAB programmings. Analyses of results obtained in MATLAB. Results compared with published work are in good

agreement. MHD is able to resolve issue involving separation of flow. Given appropriate magnetic strength, separation of flow is

no longer detected. MHD has potential to increase heat transfer at the surface

of sphere. Opposing flow requires stronger magnetic field to

encounter separation of flow to be compared to assistingflow.

22

Conclusion

M ts xs f Cf Re1/2

s NuRe1/2

(x = 0) (x = 180) (x = 0) (x = 180)

M ts xs f Cf Re1/2

h NuRe1/2

(x = 0) (x = 180) (x = 0) (x = 180)

( = 0) ( = 0, = ) (Pr = 0.7)

( ) ( ) (Pr = 7)

= 1 = 1

Pr = 0.7 lowest M highest M

Pr = 7 2nd lowest M 2nd highest M

23

Suggestions for Future Works

induced magnetic field. electric field. method to determine appropriate values of M. other geometries: blunt bodies, circular cylinder, elliptic

cylinder. other effects: Hall effect, internal heat generation,

Newtonian heating, heat flux, and much more.

24

Publication / AwardsISI Indexed Publication : N.F. Mohammad, A.R.M. Kasim, A. Ali, and S. Shafie. (2014). Separation times analysis of

unsteady magnetohydrodynamics mixed convective flow past a sphere. AIP ConferenceProceedings 1605: 349-354.

N.F. Mohammad, M. Jamaludin, A. Ali, and S. Shafie. (2012). Separation Times Analysis ofUnsteady Boundary Layer Flow Past an Elliptic Cylinder Near Rear Stagnation Point. WorldApplied Sciences Journal 17 (Special Issue of Applied Math): 27-32.

N.F. Mohammad, A.R.M. Kasim, A. Ali, and S. Shafie. (2012). Unsteady m